Q: Why wasn't v1 bundled with Arrowsmith or uploaded to Steam Workshop?
A: In order for this fix to work, the script file had to be a "loose" file (ie. in the Data\Scripts folder). Skyrim wouldn't load modified version of original game scripts from BSA files, but it would load them if they are loose. And Steam Workshop doesn't allow loose files. However, Skyrim does appear to load modified versions of original scripts from the BSA now, so either a patch changed it or I was just doing it wrong before.
Arrowsmith v.3c now has the fix built in.

Q: What caused the bug?
A: Hawks are moving objects rather than normal NPCs. The script disables them when they get hit, places a lootable "hawk" container at their location (which falls to the ground), then re-enables the moving object after 72 in-game hours to "respawn" them. For some reason the developers had this set to only happen once for each hawk. So when they respawned, nothing would happen when they got hit. If you tried to shoot them with arrows, the arrows would just stick to them and they would keep flying.